Joseph Kouneiher

The p-adic conception of number, inaugurated by Kurt Hensel in 1897, redefined the meaning of completeness and proximity by replacing magnitude with divisibility. Born within number theory, it has since transformed the conceptual foundations of mathematics and physics alike. This paper traces the historical genesis and philosophical significance of this transformation-from the arithmetization of the continuum to the emergence of p-adic and adelic structures. Methodologically, we combine a concise mathematical exposition of p-adic and adelic constructions with a structural analysis of their role in modern mathematical physics, particularly in renormalization and scale hierarchies. We show how ultrametric topology and valuation naturally encode hierarchical regularity and yield convergent structures where Archimedean frameworks encounter divergence. Conceptually, we argue that the p-adic and adelic frameworks do not merely extend analysis but reconfigure the categories of infinity, unity, and knowledge. Infinity appears not as boundless extension but as hierarchical depth; unity arises from coherence among distinct local worlds; and knowledge unfolds as refinement rather than approximation. Taken together, these developments lead to an arithmetic metaphysics in which reality is defined by coherence across scales—an ontology of structure and harmony rather than of substance and extension.

This article examines the symbolic epistemology and mathematical theology of Nicholas of Cusa (1401–1464), arguing that his integration of theological symbolism, speculative metaphysics, and geometrical imagination constitutes a pivotal reconfiguration of natural philosophy. Far from anticipating modern science in a methodological sense, Cusanus offers a symbolic vision of nature rooted in docta ignorantia, the coincidence of opposites, and a relational cosmology grounded in divine infinity. Through analysis of key works—including De Docta Ignorantia, De Coniecturis, and Idiota de Staticis Experimentis—we show how Cusanus’s use of mathematics functions not as predictive formalism but as contemplative ascent. His thought resists the dichotomies of faith and reason, symbol and concept, experiment and metaphysics, offering instead a unified vision of knowledge as symbolic participation in divine intelligibility. The study situates Cusanus within both his medieval context and broader philosophical lineages, including resonances with Kantian epistemology, structuralist formalism, and contemporary theologies of science. Ultimately, this paper argues that Cusanus’s symbolic vision provides valuable resources for rethinking the relationship between knowledge, theology, and mathematics. Rather than directly opposing scholastic essentialism or modern rationalism, his epistemology offers a third path—grounded in humility, approximation, and symbolic mediation—that anticipates and challenges later philosophical and theological developments.

Our aim in this paper is to replace the old concept of truth in mathematics, based on the Set Structure provided with idea of true and false characterized by the presence of a characteric function \, by a mathematical structures founded on the idea of Topos, the triple structure \\}\) and the notion of Gradual Truth or Steps from the truth. Our motivations is to understand the mathematical structures underlying the emergence’s mechanism and phenomena. We think that this approach could be useful to better appreciate the subtleties of the notion of truth and gives us a better understanding of the complex and changing phenomena.